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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 11th 2016
• (edited Jan 11th 2016)

At the old entry cohomotopy used to be a section on how it may be thought of as a special case of non-abelian cohomology. While I (still) think this is an excellent point to highlight, re-reading this old paragraph now made me feel that it was rather clumsily expressed. Therefore I have rewritten (and shortened) it, now the third paragraph of the Idea-section.

(We had had long discussion about this entry back in the days, but it must have been before we switched to nForum discussion, because on the nForum there seems to be no trace of it.)

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeJan 11th 2016

Back over here and here.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 15th 2018
• (edited Mar 15th 2018)

briefly recorded some facts (here) on cohomotopy of 4-manifolds, from Kirby-Melvin-Teichner 12

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 18th 2018

added a few words in the Properties-section on the isomorphism between cohomotopy classes of smooth manifolds and the (normally framed) cobordism group in complementary dimension: here

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 28th 2018

copied to here the remark about configuration spaces of points with labels in $S^n$ computing (twisted, unstable) cohomotopy (here)

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMay 6th 2019

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeAug 23rd 2019
• (edited Aug 23rd 2019)

added graphics (here) illustrating the unstable Pontrjagin-Thom isomorphism

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeAug 23rd 2019

added also a graphics (here) illustrating the example of $\pi^n\big( (\mathbb{R}^n)^{cpt}\big) \simeq \mathbb{Z}$ under the PT-iso

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeAug 23rd 2019

added also graphics (here) illustrating the $\mathbb{Z}_2$-equivariant version of the previous example.

Am adding the same illustration also to the respective discussion at equivariant Hopf degree theorem

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeAug 23rd 2019
• (edited Aug 23rd 2019)

[duplicate announcement deleted]

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeAug 23rd 2019

further in this sequence of examples: added graphics (here) illustrating the equivariant Cohomotopy of toroidal orientifolds

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeAug 31st 2019

added one more graphics (here), meant to illustrated how the normal framing of the submanifolds encodes the sign of the “Cohomotopy charge” which these carry, under PT

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeSep 27th 2019
• (edited Sep 27th 2019)

• H. Sati, U. Schreiber:

where those graphics are taken from

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeOct 6th 2019

added these references on Cohomotopy cocycle spaces:

• Vagn Lundsgaard Hansen, The homotopy problem for the components in the space of maps on the $n$-sphere, Quart. J. Math. Oxford Ser. (3) 25 (1974), 313-321 (DOI:10.1093/qmath/25.1.313)

• Vagn Lundsgaard Hansen, On Spaces of Maps of $n$-Manifolds Into the $n$-Sphere, Transactions of the American Mathematical Society Vol. 265, No. 1 (May, 1981), pp. 273-281 (jstor:1998494)

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeOct 20th 2019
• (edited Oct 20th 2019)

prodded by Dmitri (here) I have added a remark on terminology (here). In the course of this I ended up considerably expanding the Idea-section; now it has also a subsection “As the absolute cohomology theory” (here)

• CommentRowNumber16.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 20th 2019
Thanks! In the table of flavors of Cohomotopy, shouldn't we also have differential Cohomotopy?
• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeOct 20th 2019

Yes! Eventually we need a higher dimensional table. Or a table of tables.

Personally, of course I am eager to go full blown into twisted equivariant differential Cohomotopy of super orbifolds. And Vincent has a bunch of ideas for what to do. But to make sure not to be barking up the wrong tree, we’d first like to finish one or two further consistency checks in the “topological sector” first.

But that’s just me. If you want to go ahead creating more nLab entries on further variants, please do.

• CommentRowNumber18.
• CommentAuthorDavid_Corfield
• CommentTimeOct 23rd 2019

• Peter Franek, Marek Krčál, Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory, (slides)
• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeOct 23rd 2019

Thanks for the pointer. That made me add also the article that it’s based on:

• Martin Čadek, Marek Krčál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek, Uli Wagner, Computing all maps into a sphere, Journal of the ACM, Volume 61 Issue 3, May 2014 Article No. 1 (arxiv:1105.6257)
• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeOct 24th 2019

I think the talk is closer to

so have added that. It seems to rely on values of a function bounded away from $0$ being mapped to a sphere.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeOct 28th 2019

Thanks! Interesting. I am adding cross-links with persistent homology (in lack of a general mathematical notion of “persistency” of which these two are examples(?))

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeFeb 6th 2020

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeFeb 8th 2020

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeFeb 16th 2020

• Victor Vassiliev, Twisted homology of configuration spaces, homology of spaces of equivariant maps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials (arXiv:1809.05632)
• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeAug 4th 2020

• Robert West, Some Cohomotopy of Projective Space, Indiana University Mathematics Journal Indiana University Mathematics Journal Vol. 20, No. 9 (March, 1971), pp. 807-827 (jstor:24890146)
• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeSep 6th 2020

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeNov 25th 2020

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeNov 25th 2020

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeDec 13th 2020

• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeDec 14th 2020
• (edited Dec 14th 2020)

added this pointer, on Cohomotopy sets of Thom spaces:

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeDec 14th 2020

I have added (here) the statement of how composition in Cohomotopy corresponds, under PT, to products of submanifolds, together with a pasting diagram that shows what’s going on

• CommentRowNumber32.
• CommentAuthorDavid_Corfield
• CommentTimeDec 14th 2020

Does this composition amount to some kind of algebra structure on the cohomotopy groups of the spheres with an action?

• CommentRowNumber33.
• CommentAuthorUrs
• CommentTimeDec 16th 2020

Sure, composition of $Maps(S^n, S^{n'})$ is the unstable precursor of the graded ring structure on the stable homotopy group of spheres.

Because, by the exchange law (functoriality of the smash product) we may read the product on homotopy groups of spheres in the usual form

$S^{n_1 + n_2} \simeq S^{n_1} \wedge S^{n_2} \overset{ f_1 \wedge f_2 }{ \longrightarrow } S^{n'_1} \wedge S^{n'_2} \simeq S^{ n'_1 + n'_2 }$

equivalently as

$S^{n_1 + n_2} \simeq S^{n_1} \wedge S^{n_2} \overset{ f_1 \wedge id_{S^{n_2}} }{ \longrightarrow } S^{n'_1} \wedge S^{n_2} \overset{ id_{S^{n'_1}} \wedge f_2 }{ \longrightarrow } \simeq S^{ n'_1 + n'_2 }$

Since, after stabilization, $f_i$ is the same as $f_i \wedge id_{S^k}$, this second version exhibits the product on the homotopy groups of spheres as given by composition of representing maps.

• CommentRowNumber34.
• CommentAuthorDavid_Corfield
• CommentTimeDec 16th 2020

So I guess I’m reaching for some kind of sphere operad, as this question asks at MathOverfow.

• CommentRowNumber35.
• CommentAuthorUrs
• CommentTimeDec 16th 2020
• (edited Dec 16th 2020)

By construction, that operad of maps $\underset{i}{\prod} S^{n_i} \longrightarrow S^{n}$ (MO:q/142093) acts on Cohomotopy sets $\pi^{n_i}( X_i )$, by postcomposition of representing maps $X_i \to S^{n_i}$.

There is something slightly more interesting which plays a role for measuring brane charges (writeup upcoming…), namely Cohomotopy classes of spaces which are one-point compactifications of the form

$\big( \mathbb{R}^{b-p} \times S^{9-b} \big)^{cpt} \;\simeq\; S^{b-p} \wedge S^{9-b}_+ \;\simeq\; \big( S^{b-p} \times S^{9-b} \big)/( \{\infty\} \times S^{9-b} )$

So these are near-horizon geometries of solitonic $b$-branes in 11d with probe $p$-branes bound to them. They are not quite the plain Cartesian product of spheres, but the result of smashing down at infinite-distance from the $p$-brane the sphere around the $b$-brane.

(An illustration is currently here, but this is temporary and will disappear soon.)

• CommentRowNumber36.
• CommentAuthorUrs
• CommentTimeDec 20th 2020

• CommentRowNumber37.
• CommentAuthorUrs
• CommentTimeDec 21st 2020

• Nobuo Shimada, Homotopy classification of mappings of a 4-dimensional complex into a 2-dimensional sphere, Nagoya Math. J., Volume 5 (1953), 127-144 (euclid:nmj/1118799399)
• CommentRowNumber38.
• CommentAuthorUrs
• CommentTimeDec 23rd 2020

• CommentRowNumber39.
• CommentAuthorUrs
• CommentTimeFeb 5th 2021

I have added (here) mentioning of “reduced Cohomotopy” and statement and proof of its coincidence with unreduced Cohomotopy in positive degree.

• CommentRowNumber40.
• CommentAuthorDavid_Corfield
• CommentTimeMay 28th 2021

I hadn’t thought about the case $n=0$ before, but cohomotopy in that degree sends a space to subsets of components.

Thinking negatively, if as in arXiv:math/0608420, p. 14, $S^{-1} = \emptyset$, then $(-1)$-homotopy is as here

The negation of a proposition $\phi$, regarded as the collection of its proofs is $\phi \to \emptyset$. A cocycle $p:\phi \to \emptyset$ is a proof that $\phi$ is false.

• CommentRowNumber41.
• CommentAuthorUrs
• CommentTimeMay 28th 2021
• (edited May 28th 2021)

Yes. So this means that brane charge measured in equivariant cohomotopy $\pi^V$ sees a special kind of brane sitting at $G$-fixed poins whenever $V^G = 0$, namely a kind of brane of which there is either none or one, but which cannot be superimposed to $N \geq 2$-branes. These “special kind of branes” may be identified (p. 4 here) with O-planes – which, indeed, are supposed to be much like branes, but of which there is either one or none at a given spot.

$\,$

By the way, we submitted our proposal last night. Now I am hitting the road to cross continents in a crazy world. Tomorrow at this time I should be back in the country whose passport I carry. Then I’ll need 24 hours to recover. And then I hope to slowly get back to computing entanglement entropy of the Cayley state.

• CommentRowNumber42.
• CommentAuthorDavid_Corfield
• CommentTimeMay 30th 2021

Cohomotopy in the lowest dimensions

Since the 0-sphere is the disjoint union of two points, $0$-cohomotopy corresponds to the powerset of the connected components of a space.

Further, the $(-1)$-sphere is understood as the empty space. Since the only map to the empty space is the identity map from the empty space, the $(-1)$-cohomotopy of a space is measuring whether that space is empty. In the context of homotopy type theory, this is the same as negation.

For 0-cohomotopy should one speak of decidable subsets?

• CommentRowNumber43.
• CommentAuthorDavid_Corfield
• CommentTimeMay 30th 2021

Re #41, hope the trip back wasn’t too bad.